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Traversing a Set of Points with a Minimum Number of Turns ABSTRACT
"... Given a finite set of points S in Ê d, consider visiting the points in S with a polygonal path that makes a minimum number of turns, or equivalently, has the the minimum number of segments (links). We call this minimization problem the minimum link spanning path problem. This natural problem has app ..."
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Given a finite set of points S in Ê d, consider visiting the points in S with a polygonal path that makes a minimum number of turns, or equivalently, has the the minimum number of segments (links). We call this minimization problem the minimum link spanning path problem. This natural problem has appeared several times in the literature under different variants. The simplest one is where the allowed paths are axisaligned. Let L(S) be the minimum number of links of an axisaligned path for S denote by G d n the ddimensional grid of size n. Kranakis, Krizanc and Meertens (Ars Combinatoria, vol. 38, pp. 177–192, 1994) showed that in 2dimensions L(G 2 n)=2n − 1 and in three dimensions 4 3 n2 − O(n) ≤ L(G 3 n) ≤ 3 2 n2 + O(n). Kranakis et al. conjectured that, for all d ≥ 3, L(G d n) = d d−1 nd−1 ±O(n d−2). We prove the conjecture for d =3byshowingthatL(G 3 n) ≥ 3 2 n2 − O(n). For general d, we give new estimates on L(G d n), that bring us very close to the conjectured value. The new lower bound of (1 + 1 d)nd−1 − O(n d−2) improves previous result by Collins and Moret (Information Processing Letters, vol. 68, pp. 317–319, 1998), while the new upper bound of (1+ 1 d−1)nd−1 +O(n d−3/2) differs from the conjectured value only in the lower order terms. For arbitrary point sets, we give an exact bound on the
Path Planning Algorithms under the LinkDistance Metric
, 2006
"... The Traveling Salesman Problem and the Shortest Path Problem are famous problems in computer science which have been well studied when the objective is measured using the Euclidean distance. Here we examine these geometric problems under a different set of optimization criteria. Rather than consider ..."
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The Traveling Salesman Problem and the Shortest Path Problem are famous problems in computer science which have been well studied when the objective is measured using the Euclidean distance. Here we examine these geometric problems under a different set of optimization criteria. Rather than considering the total distance traversed by a path, this thesis looks at reducing the number of times a turn is made along that path, or equivalently, at reducing the number of straight lines in the path. Minimizing this objective value, known as the linkdistance, is useful in situations where continuing in a given direction is cheap, while turning is a relatively expensive operation. Applications exist in VLSI, robotics, wireless communications, space travel, and other fields where it is desirable to reduce the number of turns. This thesis examines rectilinear and nonrectilinear variants of the Traveling Salesman Problem under this metric. The objective of these problems is to find a path visiting a set of points which has the smallest number of bends. A 2approximation algorithm is given for the rectilinear problem, while for the nonrectilinear problem, an O(log n)approximation algorithm is given. The latter problem is also shown to be NPComplete.
Covering Paths for Planar Point Sets
, 2013
"... Given n points in the plane, a covering path is a polygonal path that visits all the points. If no three points are collinear, every covering path requires at least n/2 segments, and n−1 straight line segments obviously suffice even if the covering path is required to be noncrossing. We show that ev ..."
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Given n points in the plane, a covering path is a polygonal path that visits all the points. If no three points are collinear, every covering path requires at least n/2 segments, and n−1 straight line segments obviously suffice even if the covering path is required to be noncrossing. We show that every set of n points in the plane admits a (possibly selfcrossing) covering path consisting of n/2+O(n/logn) straight line segments. If the path is required to be noncrossing, we prove that (1−ε)n straight line segments suffice for a small constant ε> 0, and we exhibit nelement point sets that require at least 5n/9−O(1) segments in every such path. Further, the analogous question for noncrossing covering trees is considered and similar bounds are obtained. Finally, it is shown that computing a noncrossingcoveringpath for n points in the plane requires Ω(nlogn) time in the worst case. 1
The Rectilinear kBends TSP
"... Abstract. We study a hard geometric problem. Given n points in the plane and a positive integer k, the Rectilinear kBends Traveling Salesman Problem asks if there is a piecewise linear tour through the n points with at most k bends where every linesegment in the path is either horizontal or vertic ..."
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Abstract. We study a hard geometric problem. Given n points in the plane and a positive integer k, the Rectilinear kBends Traveling Salesman Problem asks if there is a piecewise linear tour through the n points with at most k bends where every linesegment in the path is either horizontal or vertical. The problem has applications in VLSI design. We prove that this problem belongs to the class FPT (fixedparameter tractable). We give an algorithm that runs in O(kn2 + k4kn) time by kernelization. We present two variations on the main result. These variations are derived from the distinction between linesegments and lines. Note that a rectilinear tour with k bends is a cover with k linesegments, and therefore a cover by lines. We derive FPTalgorithms using boundedsearchtree techniques and improve the time complexity for these variants. 1